Microscopie non-lineaire polarimetrique dans les milieux moleculaires et biologiques

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Microscopie non-lineaire polarimetrique dans les milieux moleculaires et biologiques

Alicja Gasecka

To cite this version:

Alicja Gasecka. Microscopie non-lineaire polarimetrique dans les milieux moleculaires et biologiques.

Physics [physics]. Université Paul Cézanne - Aix-Marseille III, 2010. English. �tel-00560415�

THÈSE

pour obtenir le grade de Docteur en Sciences de l'Université Paul Cézanne - Aix-Marseille III

Discipline : Optique photonique et traitement d'image

Polarimetric multiphoton uorescence microscopy in molecular and biological media.

Microscopie non-linéaire polarimétrique dans les milieux moléculaires et biologiques.

soutenue publiquement le 10 Decembre 2010 par

Alicja G¡secka

École Doctorale : Physique & Sciences de la Mati£re

Rapporteurs : Prof. Andrzej Miniewicz Prof. Francesco Pavone Examinateurs : Dr. Emmanuel Beaurepaire Directeurs de th£se : Dr. Sophie Brasselet

Prof. Joseph Zyss

Abstract

Abstract

Light-matter interaction in molecular and bio-molecular media can lead to complex pro- cesses where optical elds polarizations couple to an assembly of molecular transition dipoles. The manipulation of the optical elds polarization in uorescence microscopy can in particular give access to ne changes occurring in molecular arrangements. In this PhD thesis we report a method based on a tuneable excitation polarization state complemented by a polarized read-out, applied to polarization-resolved multiphoton u- orescence microscopy. Two-photon uorescence polarimetry allows to retrieve a quanti- tative information on the static molecular distribution shape and orientation in dierent environments such as model lipid membranes, cell membranes, and molecular inclusion compounds that can be strongly heterogeneous. Three-photon uorescence polarimetry has been furthermore applied in bio-molecular media in order to provide a diagnostics for crystallinity in protein crystals with high sensitivity to their structure and symme- try. The experimental implementation of polarimetric multi-photon microscopy requires to quantify possible polarization distortions originating from the experimental set-up or sample itself, which are thoroughly analyzed.

Keywords : multiphoton uorescence, polarization, microscopy, molecular distribution,

membranes, zeolite L, protein crystals.

Résumé

Les interactions lumière-mati£re dans les mileux moléculaires et bio-moléculaires peuvent mener ¯ des processus complexes où les polarisations des champs optiques se couplent aux assemblages de dipoles de transitions moléculaires. La manipulation des polari- sations des champs optiques en microscopie de uorescence peut en particulier donner accès à des modications nes d'arrangements moléculaires. Dans ce travail de thèse nous introduisons une méthode basée sur la variation continue d'un état de polarisation d'excitation complémentée par une analyse polarisée, appliquée à la microscopie de uo- rescence multi-photons. La uorescence à deux photons polarimétrique permet d'accéder à une information statique quantitative sur la forme et l'orientation de la distribution orien- tationnelle moléculaire dans des membranes lipidiques articielles, dans des cellules ou sur des composés molécluaires co-cristallins qui peuvent être fortement hétérogènes. La u- orescence à trois photons polarimétrique apporte de plus un diagnostique de cristallinité dans des cristaux de protéines, avec une forte sensibilité à leur structure et symétrie.

L'implémentation expérimentale de cette technique requiert de quantier les distortions de polarisation provenant du montage expérimental et de l'échantillon lui-même, qui sont nement analysés.

Mots clefs : uorescence multi-photon, polarisation, microscopie, distribution molécu-

laire, membranes, zeolite L, cristaux de protéines

Acknowledgements

Acknowledgements

It is a pleasure to thank the many people who made this thesis possible.

First and foremost I oer my sincerest gratitude to Sophie Brasselet for excellent supervision, advice, and guidance. Throughout my thesis she supported me with her patience, enthusiasm, great eorts to explain things clearly and simply, and knowledge whilst allowing me to work in my own way. Above all and the most needed, she provided me uninching encouragement by her truly scientist intuition that has made her a con- stant oasis of ideas and passions in science, which exceptionally inspire and enrich my growth as a student, a researcher and a scientist want to be. One simply could not wish for a better or friendlier supervisor.

I acknowledge Prof Joseph Zyss for the opportunity to do my PhD at the Institut Fres- nel. I would like to thank Hervé Rigneault for his very warm welcome upon my arrival at the MOSAIC group and for his invaluable advice with presentations produced over

Figure 1: Alicja G¡secka in Calanques.

the past three years.

I would like to thank the members of the jury: Prof Andrzej Miniewicz, Prof Francesco Pavone and Dr Emmanuel Beaure- paire for their time, interest in the subject of this thesis, helpful comments and insightful questions.

I highly appreciate the collaboration with Cyril Favard for his advice, patience and help with articial cell membranes prepa- ration and Tsai-Jung Han for her enthusiasm and perseverance in cell membranes measurements. Special thanks for excellent cooperation on Zeolite L inclusion compounds belong to Le- Quyenh Dieu and Dominik Brühwiler from Institute of Inor- ganic Chemistry, University of Zürich. I would like to also thank Anita Lewit Bentley from Institut Pasteur for very u- ent cooperation and fruitful discussions on protein crystallization.

My eternal gratitude goes to David Gachet who patiently answered all my questions, tracked down books and papers and generally went above and beyond in helping nd any resources which might have been helpful to me.

I greatly appreciate and wish to thank all the Technical sta at Institut Fresnel for their great assistance during this project. Particular thanks go to Servane Lenne without whose support I would not be able to go through the complexity of the French bureaucracy.

The members of the MOSAIC group have contributed immensely to my personal and

professional time in Marseille. The group has been a source of friendships as well as good advice and collaboration. I would like to express my sincere thanks for all the people who have made Marseille a very special place and made all those years an unforgettable:

Fabiana, Peter, Alla, David, Pascal, Pierre B. et Pierre G., Sophie Bru, Heykel, Stéphane, Jules and many others. I am grateful for time spent with them, our memorable apréos on Cours Julien and trips to the Calanques.

During these three years I have had the opportunity to participate in Nanomatch Project in the frame of Marie-Curie Research Training Network. I highly appreciated exchange of ideas and experiences during network conferences. I would like to express my spe- cial thanks to young researchers for never-ending discussions and unforgettable meetings:

Krisztina, Jan, Varun, Anna, Lucas, Le-Quyenh, Mark, Agnieszka, Fabio, Arantxa, Ri- carda and Julius.

I would also like to thank Gabriel, not necessarily for coming along at the right time, but for the very special person he is. And for the incredible amount of patience he had with me in the last ve months.

And last but not least I would like to thank my parents for creating an environment in

which following this path seemed so natural.

Contents

Introduction 5

1 Polarization resolved uorescence microscopy 7

1.1 Multiphoton uorescence processes . . . . 7

1.1.1 One photon uorescence . . . . 7

1.1.2 Two and three-photon uorescence . . . 14

1.2 From one molecule to an assembly of molecules . . . 17

1.3 Polarization resolved uorescence analysis . . . 23

1.3.1 Fluorescence anisotropy . . . 23

1.3.2 Polarimetric analysis : a tool to probe molecular angular distribu- tion functions . . . 27

1.3.3 Comparison of one, two and three-photon uorescence . . . 29

1.3.4 Read-out of a symmetry information using polarization resolved uorescence . . . 30

1.4 Conclusion . . . 36

2 Polarization distortion eects in polarimetric multiphoton microscopy 37 2.1 Polarization resolved experimental set-up . . . 38

2.1.1 Two-photon uorescence microscopy set-up . . . 38

2.1.2 Three-photon uorescence set-up . . . 42

2.2 Polarization distortion introduced by reection optics . . . 45

2.2.1 Inuence of dichroism and ellipticity on the polarization response . 45 2.2.2 Polarization distortion by high NA focussing . . . 52

2.2.3 Polarization distortion by high NA collection . . . 54

2.3 Eect of the absorption-emission dipoles relative angle on the polarimetric data . . . 58

2.4 Eect of the uorescence resonant energy transfer on the polarimetric data 60

2.4.1 Homo-FRET in statistic distributions . . . 61

2.4.2 Homo-FRET in crystals . . . 64

2.5 Inuence of birefringence on the polarimetric responses . . . 65

2.5.1 In situ characterization of the sample local birefringence . . . 67

2.5.2 Inuence of birefringence on 2PEF polarimetric responses . . . 69

2.6 Conclusion . . . 71

3 Probing molecular organization in biological membranes using two-photon uorescence imaging 73 3.1 The lipid membrane: an insight into cell functions . . . 73

3.1.1 Giant Unilamellar Vesicles (model cell membrane) . . . 75

3.1.2 Cell membrane . . . 78

3.1.3 Fluorescent lipid probes . . . 79

3.2 Polarimetry in lipid membranes: theoretical model . . . 83

3.2.1 Molecular distribution model . . . 83

3.2.2 Anisotropy analysis . . . 86

3.2.3 Polarimetric 2PEF microscopy : inuence of the distribution aper- ture, shape and average orientation. . . 88

3.3 Experimental results . . . 91

3.3.1 Investigation in two-phase lipid mixtures GUVs. . . 91

3.3.2 Investigation in cell membranes. . . 96

3.4 Conclusion . . . 100

4 Molecular order in molecular inclusion compounds: "crystal-type" sam- ples 103 4.1 Host-guest material compounds: zeolite L . . . 103

4.2 Zeolite L characteristics . . . 106

4.3 Fluorescent doping dyes . . . 107

4.4 Molecular distribution theoretical model . . . 110

4.4.1 Historical model . . . 110

4.4.2 Accounting for molecular disorder . . . 111

4.4.3 The eect of Θ and Ψ on the 2PEF polarimetric response: sensitiv- ity of the technique . . . 113

4.4.4 Possible distortion of the polarimetric responses due to the sample properties . . . 114

4.5 Experimental results . . . 116

4.5.1 The quality of the t . . . 120

Contents

4.6 Conclusion . . . 121

5 Polarization resolved Three-photon uorescence in crystals 123 5.1 Samples characteristics and preparation . . . 124

5.1.1 P-terphenyl crystals . . . 124

5.1.2 Lysozyme . . . 126

5.2 Modeling three-photon uorescence responses from crystals . . . 127

5.3 Analysis of experimental results . . . 132

5.4 P-terphenyl crystals . . . 132

5.5 Lysozyme crystals . . . 135

5.6 Conclusion . . . 137

Conclusion and perspectives 142

Bibliography 142

Introduction

Microscopy imaging, when combined with uorescent labeling has become an essential tool in biology, biochemistry as well as medical sciences for a variety of applications from molecular and cell biology investigations to biomedical diagnostics [1, 2]. Fluorophores attached to proteins, biomolecular assemblies or lipids have made it possible to identify cells and sub-microscopic cellular components behavior with exquisite sensitivity and se- lectivity. Localization in vitro or in vivo of specic biomolecules (proteins, lipids) in single cells or in tissues permits the investigation of their interactions with neighboring molecules as well as environment which is a prerequisite for the understanding of their biological functions. The development of uorescence microscopy would not have been possible without molecular engineering, that have lead to molecular and non-organic (semicon- ductor nanoparticles) probes able to stain biological components otherwise inaccessible for visible-range optical microscopy. These probes oer a large variety end-groups in or- der to target specic biological molecules attachments. They are also designed such as to cover a large range of absorption and emission wavelength ranges which makes uores- cence staining a very exible solution. Their signicant quantum yields permits to study complex biological phenomena in even delicate conditions such as tissues where autou- orescence would otherwise hamper any kind of specic imaging. Besides, the multiple uorescence labeling that uses dierent probes can simultaneously identify several target molecules. A part from these synthesized probes, the discovery of uorescent proteins has revolutionized the uorescence microscopy imaging. Being directly expressed in cells (obtained for instance by gene manipulation) these uorescent labels avoid diculties of in situ chemical staining. Among other things, green uorescent protein and its mutants, allowed comparing cancer cells with specic genotype or phenotype as well as visualiz- ing tumor cell mobility, invasion or angiogenesis [3]. Labeling single cells in the nervous system permitted to image (determining events in) neuropathology [4]. We can thus see that the molecular imaging has a great impact on the evolution of knowledge in biology and medicine.

While uorescence microscopy imaging has been primarily developed thanks to confo-

cal one-photon microscopy, other instrumentation developments have emerged in the last decades. Among the most successful ones, multiphoton microscopy initiated by Denk and colleagues in 1990 in a paper on two-photon laser scanning uorescence microscopy [5, 6]

brought a new opportunities to the study of biological matter. The concept of two-photon excitation was described by Maria Goeppert-Mayer in her doctoral thesis on the theory of two-photon quantum transitions in atoms in 1931 [7], and was experimentally observed af- ter the development of laser sources, in caesium vapor in 1962 [8]. The higher order three- photon uorescence was rst demonstrated in 1964 with 20 ns pulse ruby laser [9] and high quality uorescent images generated by three-photon excitation were then obtained in 1996 by laser-scanning microscopy [10, 11]. The interest in multiphoton microscopy comes from the fact that multiphoton excitation has intrinsic advantages relative to the one photon uorescence process. First, the use of infrared excitation light leads to less scattering, and therefore deeper optical penetration which is crucial for tissues imaging.

Secondly, the non linear nature of the excitation leads to an intrinsic spatial resolution which removes the need of confocal detection schemes. It furthermore reduces the angular photoselection and thus ameliorates the angular sensitivity of polarized measurements as will be detailed in the present work. Moreover reduction in phototoxic eects in some cases makes multiphoton excitation imaging an attractive method for imaging uorescent probes in vivo. The combination of two and three-photon excitation nally extends the capabilities of a multiphoton imaging system since a single wavelength can provide local- ized excitation of a wide variety of uorophores.

Two-photon uorescence imaging for biology is applied to two main topics : cell mem-

brane imaging and tissues imaging. Cell and model cell membrane studies, using about

the same lipid probes as for one-photon uorescence, have essentially focused on cell ar-

chitecture imaging by probing local polarity [12] and local environment [13]. Two-photon

uorescence analysis has been completed with second harmonic generation (SHG) imag-

ing microscopy (which is a coherent process occurring in non-centrosymmetrical media)

to provide more information on molecules present in an outer leaet of the membrane,

pointing outside [14]. Tissues imaging does not necessarily require the use of exogenous

stains since these studies are performed on intrinsic autouorescence indicators such as

numerous uorescent proteins such as avins or elastin [15, 16, 17]. Information on tis-

sue morphology, cell behavior or diseases states is often obtained by combining dierent

contrasts for instance multiphoton uorescence with coherent microscopies : Second Har-

monic Generation (SHG) [18, 19, 20], Third Harmonic Generation (THG) [21, 22] or

Coherent Anti-Stokes Raman Scattering (CARS) [23, 24]. At last, three-photon uores-

Introduction

cence for biology has been reported in a few works that aimed to image a xed and living biological specimens such as embryos and cells stained with uorescent labels [25, 26].

These studies however are more delicate compared to two-photon uorescence imaging due to the lower eciency cross-sections of the uorescent proteins in this regime.

An issue that has an essential importance in a large variety of biological phenom- ena is molecular order, or the way molecules are oriented in an ensemble. Measur- ing orientational information is essential to understand the molecular interactions that drive the morphology of biomolecular assemblies, from membrane proteins aggregates to biopolymers involved in signaling events [27, 28, 29], cell mechanics and adhesion [30, 31].

However imaging such organizations using optical microscopy remains a challenge, which has been essentially approached by uorescence anisotropy measurements from molecular probes localized at adequate functional positions. This technique is based on measur- ing uorescence polarization response ratio of target molecules in a sample using two excitation/detection polarization directions and can be applied for one-photon or mul- tiphoton uorescence. A linearly polarized incident electric eld preferentially excites uorescent target molecules with transition dipole moments aligned parallel to the inci- dent polarization vector, therefore providing orientation sensitivity. Such a scheme has been widely used either in time-resolved [32, 33] or steady state schemes in order to an- swer conformational and structural questions in isotropic environments where molecules are orientationally averaged. Steady state uorescence anisotropy can be readily imple- mented in imaging, in particular in membranes where uorescent probes can be inserted.

For example anisotropy analysis has been used to obtain an information on the orienta- tion of long acyl chain uorescent carbocyanine dye transition dipoles and on the dyes rate of rotation in a biological membrane [34], as well as local orientational distribution of actin laments within a cell [35]. The use of uorescent lipid probes allowed to get insight into the membrane lipid organization by studying phospholipid molecular motion [36]. It was shown that two-photon uorescence anisotropy imaging can accurately image lipid organization in cell membranes and in ordered structures such as membrane nanotubes connecting two cells [13]. Studies on membrane ruing of natural killer cell immune synapses have also been recently published [29]. These membrane studies have been extended to proteins order investigation, however with more diculty since studying a protein orientation requires rigidly labeled uorophores. Investigations on membrane pro- tein receptors order [37] and on the averaged orientation angle of protein laments have been nevertheless successfully implemented [38].

Fluorescence anisotropy is however limited in terms of accessible information since it

only uses two parameters of information. Fluorescence anisotropy imaging has been there-

fore successful in specic cases which are limited to only geometries of the molecular an- gular distributions which are of cylindrical symmetry and well dened orientation [35, 13], or from which only an average molecular orientation angle is retrieved [38]. Deciphering molecular and biomolecular behaviors however generally requires dealing with complex molecular angular distributions that can strongly dier from purely cylindrical symmetry distributions. Studying molecular order would also strongly benet from techniques that do not require the a priori knowledge of the mean molecular orientation. This is indeed a limitation in the current measurements in cell membranes mentioned above, in which either the contour has to be determined before any polarization resolved analysis [13] or for which studies are limited to cells of round simple shapes. For these reasons more rened polarization-dependent analysis is required. The goal of this work is to explore the capabilities of polarization-resolved two-photon uorescence in typical studies in bi- ology and material engineering. To do so, we follow a similar approach as in polarization resolved imaging in scattering media: using multiple states polarization analysis [39].

Recent works in polarization resolved coherent two-photon excitation microscopy (us- ing SHG) have demonstrated that rich information is contained in polarization responses recorded from a tuning of the incident linear polarization in the sample plane in molecular media [40, 41, 42, 43]. The outcome of these studies shows that polarization tuning gives access to ne changes of a molecular distribution that would not be accessible in a tradi- tional uorescence anisotropy measurement, as the analysis now relies on the observation of the shape of the polarization dependence response. In particular, Second Harmonic Generation polarization resolved studies have been proposed to provide information on symmetry orders in complex orientational distributions in organized molecular organic media, for instance to distinguish specically and locally the nature (symmetry, disorder) of molecular assemblies in molecular monolayers [42] and in crystals down to the nano- metric scale [43]. Polarization-controlled contrast improvement schemes have been also applied to a variety of contrasts in nonlinear coherent imaging for biological and chemistry applications, such as Second Harmonic Generation from doped membranes under electric elds [14] and collagen [44, 45, 46, 47], Third Harmonic Generation [48, 49], and Coher- ent Anti-Stokes Raman Scattering in tissues [50] and in crystals [51]. The polarimetric approach opens thus a new scope of structural studies in biological and molecular media.

While a lot of work has been done in coherent nonlinear microscopy, less work exist on

multiphoton uorescence [40]. Our goal is to focus on two and three-photon uorescence

contrasts which are of great interest in microscopy imaging today. We attempt to demon-

strate the feasibilities of Polarimetric Two-Photon Excitation Fluorescence Microscopy by

Introduction

investigation of complex molecular organization in molecular and biological media as well as to extend the polarization resolved analysis to three-photon uorescence microscopy to provide a new tool for microscopy structural and molecular order imaging.

Chapter 1 of this thesis recalls the principles of the multiphoton processes applied in uorescence microscopy and describes the context of polarization resolved analysis. The basic formalism is introduced and the theoretical approach allowing for modeling uores- cence processes in ordered media is explained in details.

In order to provide relevant information on the studied system, polarimetric analysis necessitates accounting for possible polarization distortions occurring in the experimental set-up or originating from the sample itself. Chapter 2 is devoted to this issue.

Chapter 3 presents the potential of the two-photon uorescence microscopy when probing molecular organization in biological membranes. The molecular orientation and distribution is determined in dierent lipid environments and compared with cell mem- branes.

In this introduction we have mentioned the biological context of nonlinear microscopy imaging. In chapter 4 we will show that polarization resolved uorescence microscopy can be interesting as well for materials development such as inclusion compounds made of nanochannels doped with uorescent molecules.

Finally, in chapter 5 the polarimetric analysis is extended to three-photon uorescence

microscopy. This new approach allows to distinguish between crystalline and isotropic

structures in media of high order symmetries. We present rst explorations on the polar-

ization resolved three-photon uorescence from molecular and protein crystals.

Chapter 1

Polarization resolved uorescence microscopy

Polarimetric uorescence microscopy, which is the core of this work, relies on a variation of the incident electric eld polarization to read-out light matter interaction information.

To understand the output of such experiment in terms of molecular order information one needs to detail the polarization resolved uorescence process as well as dene the notion of "molecular-order". In chapter 1 we will describe the properties of one-, two- and three-photon uorescence processes, which will be used as a contrasts in the experiments performed in this work. We will also introduce a general molecular order model that will be exploited in biological and molecular samples and detail how polarization resolved experiments can reveal information on this molecular order. Finally we will analyze the range of application of the uorescence polarization resolved technique depending on the contrast used, as well as its advantage comparing to the more traditional uorescence anisotropy technique. Essentially, we will show how to reach higher order symmetries information with multiphoton uorescence contrasts. The concepts introduced in this chapter will be used in all the rest of this work.

1.1 Multiphoton uorescence processes

1.1.1 One photon uorescence

The process by which an excited material emits light is called luminescence and among

all possible excitation processes, luminescence caused by the electromagnetic radiation

is called uorescence. One of the rst experiment showing the uorescence eect was

reported by Sir J.F.W. Herschel in 1845 [52]. He observed that a solution of quinine in water illuminated with sun emits a blue color which appears stronger when observed at a right angle relative to the direction of sunlight. A few years later, in 1852, Sir G.G. Stokes published a paper where he presented the studies on the same compound reporting that the emitted light exhibits a longer wavelength than the exciting light [53]. The historical experiment presenting this fundamental property of uorescence is shown in Fig. 1.1.

G.G. Stokes showed that while the incident blue light below 400nm is absorbed by the

Sun

Emission filter >400nm (yellow glass of wine) Excitation filter <400nm

(blue glass from church window)

Solution of quinine

G.G. Stokes

Figure 1.1: Experimental set-up used by G.G. Stokes.

quinine molecules, the emitted light is shifted to longer wavelengths (450nm), and thus can be observed by eye. The yellow (wine) lter prevents the incident radiation from reaching the observer.

The photophysical process reported by Sir J.F.W. Herschel and Sir G.G. Stokes was later understood by A. Jabªo«ski in 1935 in a model which is widely used today by spectroscopists [54]. In this model, supported by quantum chemistry, the molecule after absorption of a photon is able to undergo a radiative decay from a uorescent level reached after a fast internal conversion. This is summarized in the simplied Jabªo«ski diagram depicted in Fig.. 1.2 a. Note that this version of the diagram does not represent all the possible non-radiative decays in the relaxation process, due to the intermolecular or intra- molecular processes. In particular, intersystem crossing (form singlet to tripled states) that can result in a delayed uorescence or phosphorescence which will not be considered in the rest of this work.

The detailed excitation/relaxation process can be summarized as follows. In the

ground state | S

⟩ a molecule called uorophore or uorescent dye (generally poly-aromatic

1.1. Multiphoton uorescence processes

0 1 2 0 1 2 0 1 2

Absorption

Internal conversion

Fluorescence hn_F

hn_A

a)

Energy

core distance R₀ R₁

S₀ S₁

0 1 2

0 1 2

b)

S₀ S₁ S2

Figure 1.2: (a) Simplied Jabªo«ski diagram of one-photon uorescence process. | S

⟩,| S

⟩ represent singlet electronic levels, 0, 1, 2 are vibrational levels. (b) Illustration of the Franck-Condon principle with potential well model (molecular states as in (a)). The electronic transition is most likely to occur without changes in the positions of the nu- clei ( R

, R

- equilibrium core distances for ground state and rst excited singlet state respectively).

hydrocarbon or heterocyclic compound) lies in its lowest vibrational level 0 whereas higher

vibrational levels 1 and 2 are in general not populated (less that 1% according to Boltz-

mann statistics). The transition to higher electronic levels occurs when this molecule

absorbs a quantum of light hν

[55]. In general during excitation atom bonds get weaker,

therefore the equilibrium distance between atom cores of the molecule in excited state is

slightly larger than the one in the ground state (Fig. 1.2 b). The Franck-Condon principle

states that the excitation processes are much faster (10

s) than the time scale of nuclear

motions (10

s) (due to the lower mass of electrons as compared to nuclei), thus they do

not displace nuclei signicantly [56, 57]. Because the electronic transitions are essentially

instantaneous compared with nuclear motions the uorophore is usually excited to some

higher vibrational levels of | S

⟩ (or higher energy levels | S

⟩ depending on the incident

wavelength) which correspond to the minimal change of the nuclear coordinates. Hence

this transitions can be drawn as a vertical line on the potential curve diagram (Fig. 1.2

b). In the next 10

s , the molecule relaxes to the lowest vibrational level of | S

⟩ by

internal conversion usually triggered by molecular relaxation inuenced by solvent inter-

actions. In the most common organic uorophores the emission process occurs after a

few 10

s and originates from the lowest vibrational level of S

(this is the Kasha's rule which states that the emission level is the most stable one and does not depend on the excitation energy) [58]. Due to possible non-radiative relaxation processes (i.e. energy transfer, collisional quenching or intersystem crossing) the energy of the excited state is partially dissipated and not all the molecules initially excited by absorption return to the ground state | S

⟩ by uorescence emission. This is the reason why the uorescent eciency quantum yields are generally below 1 (this quantum yield is dened by the ratio between the radiative and the totality of the decay rates present in the system). In the last step, the uorophore comes back to the lowest vibrational level of the ground state directly by emission or by additional vibrational relaxation. Energy losses between exci- tation and emission as well as vibrational relaxations are the reason of the Stokes shift (the energy dierence between the emission spectrum and the excitation wavelength).

We will detail below the calculation of the uorescence intensity of a single molecule.

To do so, we will need to separate the steps of absorption and emission since these two steps are uncorrelated in time. Therefore we will write the emitted uorescence intensity as a product of two probabilities : the absorption between the ground and the excited state (of one, two or three photons) and the emission from the uorescent state to the ground state (of one photon).

I

∝ P

· P

. (1.1) The proportionality coecient contains collection eciencies and normalization factors.

Since our analysis does not depend on these coecients we will omit them in the future and write equals sign.

The absorption probability is calculated using quantum mechanics perturbation theory which is generally done for a one-electron atom but can be applied to a molecule [59]. We will describe here the expression for the one photon absorption probability which will be extended to two and three photons afterwards.

In the quantum mechanical picture an electromagnetic wave is considered as a pertur- bation V ˆ of the initial system, representing the energy of interaction of the atom/molecule with the externally applied radiation eld E of a frequency ω : V ˆ = − µ ˆ · (E

+E

) , where µ ˆ = − eˆ r is the electric dipole moment operator and − e is the charge of the electron at position ˆ r . The total Hamiltonian of the system H ˆ is thus written as :

H ˆ = ˆ H

+ ˆ V (1.2)

1.1. Multiphoton uorescence processes

where H ˆ

denotes the Hamiltonian for a free system without perturbation. In order to express the excitation of the molecule as a perturbation one has to use the density matrix formalism, which permits to account for the statistical environment of the molecule. Two states of the molecule given by | n ⟩ and ⟨ m | are described by the density matrix ρ ˆ elements ρ

= | n ⟩⟨ m |. The diagonal element of this matrix ρ

gives the probability that the system is in its energy eigenstate E

that corresponds to the statistical population of | n ⟩, while the out of diagonal elements ρ

quantify the coherence between levels | n ⟩ and ⟨ m |.

The Schrödinger equation written in the density matrix formalism describes how the ρ

element evolves in time:

˙

ρ

= − i

~ [ ˆ H, ρ] ˆ

− γ

(ρ

− ρ

) (1.3) where γ

is a phenomenological damping term introduced to account for dissipative interactions of the molecule with its environment, which indicates that ρ

relaxes to its equilibrium value ρ

at the rate γ

. By introducing the transition frequency ω

=

En−Em

~

between the | n ⟩ and ⟨ m | states that correspond to the solution of the unperturbed Hamiltonian, the above equation is written:

˙

ρ

= − iω

ρ

− i

~ [ ˆ V , ρ] ˆ

− γ

(ρ

− ρ

) (1.4)

˙

ρ

= − iω

ρ

− i

~

∑

ν

(V

ρ

− V

ρ

) − γ

(ρ

− ρ

) (1.5) where ∑

ν

is a sum over all intermediate states | ν ⟩. These equations cannot be solved analytically, therefore a solution has to be seek in the form of perturbation expansion performed by successive increase of the rank of V

[59]. The zero and j > 0 orders stationary solutions can be written :

˙

ρ

= − iω

ρ

− γ

(ρ

− ρ

) (1.6)

˙

ρ

= − iω

ρ

− i

~

∑

ν

( µ

ρ

− µ

ρ

)

· (E

+ E

) − γ

ρ

. (1.7) In the absence of the external eld the steady state solution of equation 1.6 is:

ρ

= ρ

. (1.8)

At thermal equilibrium, the excited states of the system may contain populations, but coherence terms of ρ

vanish. This is because thermal excitation (which is an incoherent process) cannot produce any coherent superposition of states:

ρ

= 0 for n ̸ = m. (1.9)

Using this zero solution in equation 1.7 leads to the rst order perturbation solution : ρ

(t) = ρ

− ρ

~

[ (µ

· E

)e

(ω

− ω) − iγ

+ (µ

· E

)e

(ω

+ ω) − iγ

]

. (1.10)

This term allows calculating the linear susceptibility of the molecule, using the expectation value of the induced dipole moment :

⟨ µ(t) ˆ ⟩ = T r(ˆ ρ

· µ) = ∑

n,m

µ

ρ

(t) (1.11) Under a single-frequency ω excitation (monochromatic wave) this dipole is a ω frequency oscillating term :

⟨ µ(t) ˆ ⟩ = µ

e

. (1.12)

The linear susceptibility tensor is dened by

µ

= ε

α( − ω, ω) : E

(1.13)

α

( − ω, ω) = ∑

n,m

ρ

− ρ

ε

~

µ

µ

(ω

− ω) − iγ

(1.14) when a molecule is initially in the ground state (ρ

= δ

) :

α

( − ω, ω) = 1 ε

~

∑

n

µ

µ

(ω

− ω) − iγ

. (1.15) It has been shown that the absorption probability of one photon at the frequency ω can be written as [59] :

P

= Im(α( − ω, ω)) • (E

⊗ E

) (1.16) where Im(α( − ω, ω)) is the imaginary part of the linear susceptibility of the molecule given by equations 1.14 and 1.15. In this expression ⊗ denotes the tensorial scalar product ( (E

⊗ E

)

= E

E

), and • is the tensorial product (similar to the scalar product for vectors). The demonstration of this expression requires to go to the second rank perturbation : P

is equal to the term ρ

− ρ

as an interaction of the system with the elds E

and E

[60].

In this work we will consider two level systems where only n = 1 is the dominant contribution in eq 1.15. In this case the absorption probability can be simplied in :

P

∝ α

E

E

∝ ∑

ij

µ

µ

E

E

= | µ

· E

|

(1.17)

1.1. Multiphoton uorescence processes

where the molecular absorption dipole µ

is the transition dipole moment from the ground state to the excited state. We also recognize here the transition rate between the | 0 ⟩ and

| 1 ⟩ levels in atom optics governed by the Fermi golden rule [59] : R

(ω) = 2π

~

|⟨ 0 | V ˆ | 1 ⟩|

ρ

(ω) = 2π

~

|⟨ 0 | µ · E

| 1 ⟩|

ρ

(ω) (1.18) where ρ

(ω) is the density of states of the excited level measured at the vicinity of the ω

transition frequency ( ρ

(ω) is the Lorentzian function of eq 1.15). The absorption cross section is dened by R

(ω) = σ

(ω)I

where I = 2nε

c | E

|

. Therefore :

σ

= π nε

c

1

~

| µ

|

ρ

(ω) (1.19) Typical values for the absorption cross sections of organic molecules are around 10

cm

at the maximum of the absorption peak ω = ω

(taking ρ

(ω

) =

01

) [59]. µ

will be denoted µ

in the rest of this work.

The emission probability along the analysis axis i , denoted P

, corresponds to the uorescence radiated intensity I

:

P

= I

= | E

· u

|

(1.20) where u

is a normalized vector along the direction i . The far eld E

radiated by the emission dipole µ

(which denes the transition dipole moment between the uorescent and the ground state) in the propagation direction k is expressed as [60]:

E

∝ k × (k × µ

) (1.21)

In general µ

is dierent from the absorption transition dipole that we called above µ

, due to the fact that dierent states are involved in the absorption/emission pro- cesses. This issue will be discuss in chapter 2.

Finally the one-photon uorescence intensity in the analyzing direction i is written:

I

∝ | µ

· E

|

| E

· u

|

(1.22)

The proportionality sign in Eq. 1.22 contains collection eciency factors, the uorescence

quantum yield and the proportionality coecients of the above equations.

h n

h n

h n

h n

h n

A

h n

h n

a) b)

Figure 1.3: Two and three-photon uorescence represented by a Jabo«ski diagram.

1.1.2 Two and three-photon uorescence

Multiphoton Fluorescence Microscopy relies on the quasi-simultaneous absorption of two or more photons by a molecule (Fig. 1.3). The two-photon uorescence process was pre- dicted by M. Goeppert-Mayer in 1931 [7]. During the absorption process, an electron of the molecule is transferred to an excited-state molecular orbital. The virtual absorption of a photon of non-resonant energy lasts only for a very short period (10

− 10

s) . During this time a second photon must be absorbed to reach an excited state. Whereas the selection rules for one and multi-photon absorption are dierent because of the dier- ent numbers of energy levels (virtual or not), the emission occurs from the same excited level | S

⟩. Hence the excitation spectra coming from one and multi-photon processes are not the same. This eect is clearly visible for centrosymmetric molecules where electronic levels for two-photon have generally higher energy than for one-photon excitation (indeed a two-photon absorption level has to be of dierent parity than a one-photon absorption level). Therefore an optimal wavelength for one-photon excitation is not necessarily equal to double of the wavelength for which two-photon excitation is maximal. The molecule in the excited state has a high probability to emit a photon during relaxation to the ground state. As in a one-photon excitation process, due to radiationless relaxation in vibrational levels, the energy of the emitted photon is lower compared to the sum of the energy of the absorbed photons.

In order to calculate the two-photon absorption probability P

, a similar treatment

can be applied as in the previous section, using one more order of perturbation. This prob-

ability is governed by the third order nonlinear hyperpolarizability tensor γ( − ω, − ω, ω, ω)

1.1. Multiphoton uorescence processes

[59]:

γ

( − ω, − ω, ω, ω) = 1 8ε

~

∑

n,m,ν

{ P

[µ

µ

µ

µ

]

[(ω

− ω) − iγ

][(ω

− 2ω) − iγ

][(ω

− ω) − iγ

] + P

[µ

µ

µ

µ

]

[(ω

− ω) − iγ

][(ω

− 2ω) − iγ

][(ω

− ω) − iγ

] } (1.23) with [60]:

P

= Im(γ( − ω, ω, ω, − ω)) • (E

⊗ E

⊗ E

⊗ E

) (1.24) where (E

⊗ E

⊗ E

⊗ E

)

= E

E

E

E

.

In the two level model approximation : P

∝ γ

E

E

E

E

∝ ∑

n

∑

ijkl

µ

µ

µ

µ

E

E

E

E

(1.25) the quantities µ

= µ

involve additional | n ⟩ levels in the system.

The two-photon absorption cross section calculation can also be made using the two- photon transition rate:

R

(ω) = 2π

~

∑

n

⟨ 0 | µ · E

| n ⟩⟨ n | µ · E

| 1 ⟩ (ω

− ω)

2

ρ

(2ω) (1.26)

Similarly as for one-photon process, the two-photon absorption cross section is dened by R

(ω) = σ

(ω) ¯ I

where I ¯ =

| E

|

and therefore [59]:

σ

= ω

4n

ε

c

∑

n

1

~

µ

µ

(ω

− ω)

2

2πρ

(2ω) (1.27)

Assuming that only one of the | n ⟩ levels is of dominant transition dipole and that is furthermore nonresonant at ω ( ω

− ω ≈ ω ), σ

can be estimated. Typical values around 10

m

sec/photon

can be found [11]. We can see that in the continuous excitation regime, this would lead to very low absorption eciencies comparing to the one-photon process. This is why most of the two-photon excitation studies are performed in the pulsed regime which concentrate high energy in a short time and at high repetition rate.

[61].

In the future we will mostly study one dimensional molecules in which we can assimilate µ

and µ

to a single vector direction µ

along the molecular axis. This vector µ

will allow us dening the orientation of the molecule. The two-photon absorption can thus be dened by :

P

∝ | µ

· E

|

(1.28)

Because the emission occurs from the same level as in one-photon uorescence the two- photon uorescence intensity along the analyzing direction i (expressed by the unit vector in that direction u

) is written as before :

I

∝ | µ

· E

|

| E

· u

|

(1.29) In all future calculations we will replace the ∝ by = since only the intensity dependance with respect to the incident polarization will be investigated.

Two-photon excitation exhibits several advantages over one photon uorescence in mi- croscopy. In a high peak-power pulsed laser (with mean power levels moderated to do not damage the specimen), the photon density at the point of focus is suciently high so that two photons can be simultaneously absorbed by the uorophore. Eq 1.29 shows that two-photon uorescence depends of the square of the incident intensity, therefore this process takes place only at the focus point of the microscope objective, eliminating out-of-focus excitation of a uorophore and thereby enabling 3D optical sectioning with high spatial resolution [6]. Although the axial resolution along the propagation direction is improved, it has been shown that the lateral two-photon spatial resolution is compara- ble to the one-photon confocal resolution when exciting the same uorophore [62, 63, 64].

This is due to the larger diraction-limited spot at the longer wavelength two-photon excitation source. For a 1.25 NA objective using an excitation wavelength of 960nm the typical point spread function has an FWHM of 300nm in the lateral direction and 900nm in the axial direction (which is about half the axial resolution at one-photon) [65]. Note however that this is the resolution at the surface of the sample, but it typically degrades with depth in thick scattering samples. Another advantage over the one photon process is reduced scattering, which is a major contributor to image detioration. Indeed, because the elastic scattering of light is proportional to the inverse power of the wavelength ( 1/λ

), this process is less pronounced in 2PEF and allows imaging in three times deeper regions compared to 1PEF [6]. These properties have triggered the large interest for 2PEF for bio-imaging applications.

The accessible wavelengths in widely used femtosecond Ti:Sa laser (typically 700- 1000nm) allow covering a large range of either synthesized uorophores or naturally uo- rescent proteins such as avins or elastin present in cells and tissues [6, 44, 11, 66, 67, 17].

However the majority of proteins is not maximally absorbing in the visible range

but rather in the UV (250-280nm) which is the absorption range of amino-acids. Such

wavelengths are not accessible by a two-photon excitation. In this work we are interested

1.2. From one molecule to an assembly of molecules

in targeting the uorescence from un-labeled proteins as a possible contrast for nonlinear structural imaging. For this purpose we investigated also three-photon uorescence which is a possible way to excite such UV transitions using a Ti:Sa laser.

Three photon uorescence process occurs in a manner similar to two-photon excitation.

The dierence is that three photons must be absorbed simultaneously. Following a similar approach as developed above, the three-photon absorption probability can be written using a three-photon absorption tensor :

P

∝ ∑

ijklmn

ξ

E

E

E

E

E

E

(1.30) The three-photon uorescence intensity will be dened by :

I

∝ | µ

· E

|

| E

· u

|

. (1.31) Because excitation levels are dependent on the cube of the excitation power, a spatial connement at the point of the focus is stronger, so that a higher contrast in imaging is expected. Furthermore, the resolution achieved in three-photon uorescence microscopy is greater than for one or two-photon microscopy (200nm axial and 500nm lateral resolu- tions have been theoretically predicted for a 900nm excitation wavelength and a 1.35 NA objective [10]).

1.2 From one molecule to an assembly of molecules

So far we have shown the detailed analysis of uorescence processes from a single molecule.

However in a standard microscopy measurement a great number of uorophores is present within the focal volume. Hence, in the presented section, we will extend this approach to the calculation of the uorescence signal from an assembly of molecules and discuss orientation distribution functions dening this assembly. We will focus on two-photon uorescence since 2PEF microscopy is applied to the major part of experiments performed in this thesis.

Let us consider the incident electric eld E(r) which interacts at location r with the absorption dipole moment of a molecule µ

(Ω, r) whose orientation is given by the Euler set of angles Ω = (θ, ϕ) as indicated in Fig. 1.4. The far eld E

is radiated by the emission dipole in the propagation direction k with E

(Ω, r, k) ∝ k × (k × µ

(Ω, r)) . The two-photon uorescence signal detected from one molecule in the analyzing direction i and in the direction k can therefore be expressed :

I

(Ω, r, k) = | µ

(Ω, r) · E(r) |

| E

(Ω, r, k) · u

|

(1.32)

r

θ

φ

µ

( θ , φ ,r)

µ

Y Z

X

k E

E(r)

Figure 1.4: Absorption dipole moment µ

at position r and orientation θ, ϕ excited by the incident eld E(r) in the focal volume X, Y, Z . The far eld E

is radiated by the emission dipole µ

in the propagation direction k .

As it was shown previously the uorescence emission process is completely uncorre- lated from the excitation process. This is because the absorption (and the subsequent internal conversion) plus the emission (after a stochastically distributed dwell time in the excited state) lead to a complete memory loss of the excitation conditions. Due to this incoherent nature of the uorescence process, the emission of one uorophore will be uncorrelated in time from the emission of another uorophore. Therefore the 2PEF intensity detected from an assembly of molecules is expressed as the superposition of all its individual intensity contributions. The molecular angular distribution is described by the normalized orientation distribution function f(Ω) , which is dened by the number of molecules N (Ω) dΩ = N f(Ω) dΩ present in the unit solid angle dΩ . Consequently :

I

= N

∫ ∫ ∫

| µ

(Ω, r) · E(r) |

| E

(Ω, r, k) · u

|

f (Ω) dΩ dr dk (1.33) a few remarks can be done on this expression :

• In the rigorous way the above expression (and in particular f (Ω) ) should be de-

pendent on time for molecules whose orientation is uctuating [68]. Here we will

average this quantity and give a static information within the integration time of the

measurements (typically >ms), which is much longer than the rotational diusion

time (ps-ns) of molecules and their excitation state life time (typically ns).

1.2. From one molecule to an assembly of molecules

• The incident radiation is usually focused into a nonlinear medium by the high nu- merical aperture objective in order to increase its intensity and thereby increase the eciency of the nonlinear optical process. Therefore E(r) exhibits a spatial depen- dence, which will be discussed in chapter 2. In our rst analysis, we will use the planar wave approximation which is sucient to study polarization dependencies.

• In principle E

(Ω, r, k) should be integrated over all the directions k of the high numerical aperture detection. The eect of the strong aperture on the polarization response will be studied in chapter 2 and taken into account in all the experimental analysis presented in this work. In this section for the sake of simplicity we will consider only a planar wave approximation and omit the k dependence.

• We will consider a homogeneous spatial distribution of transition dipole moments therefore the r dependance of µ

(r) will be omitted. All the possible spatial orientations will be contained in the f(Ω) function.

• The uorescence intensity scales with the number of uorophores in the focal volume, which is expected from the incoherent nature of this process. We will omit the N factor in the future.

• We assume in a rst approximation that the absorption dipole moment of a molecule µ

is parallel to its emission dipole moment µ

(non parallel dipoles will be studied in chapter 2).

With all the approximations mentioned above, the two-photon uorescence signal dened in equation 1.32, measured from a molecular assembly within an orientational distribu- tion f (Ω) and analyzed along a given polarization direction i = X, Y for an incident eld propagating along Z is now expressed:

I

=

∫

| µ(Ω) · E |

| µ

(Ω) |

f(Ω) dΩ. (1.34)

where dΩ = sin θ dθ dϕ and with θ ∈ [0 − π], ϕ ∈ [0 − 2π] . Equation 1.34 nally shows

that two-photon excited uorescence depends on both the molecular statistical orienta-

tional distribution and the excitation elds polarization. Since this eld is dened in the

macroscopic frame X, Y, Z , we need to express the components of the excitation/emission

transition dipole moments µ

(θ, ϕ) in this frame. To do so, the components of µ are

given by the polar and azimuthal angles θ and ϕ in the microscopic frame (x

, y

, z

) being

dened by the symmetry axis of the molecular angular distribution function f (Ω) (see